Let ${X}$ be a set. Let ${\mathcal{G}}$ be a non-empty collection of subsets of ${X}$ such that ${\mathcal{G}}$ is closed under finite intersections. Assume that there exists a sequence ${X_h \in \mathcal{G}}$ such that ${X = \cup_h X_h}$. Let ${\mathcal{M}}$ be the smallest collection of susbsets of ${X}$ containing ${\mathcal{G}}$ such that the following are true:

If ${E_h \in \mathcal{M}}$ ${\forall h \in \mathbb{N}}$ and ${E_h}$ ${\uparrow}$ ${E}$ then ${E \in \mathcal{M}}$
If ${E}$, ${F}$, ${E \cup F \in \mathcal {M}}$ then ${E \cap F \in \mathcal {M}}$
If ${E \in \mathcal {M}}$ then ${E^c \in \mathcal{M}}$

Does ${X \in \mathcal{M}}$ ?